Optimal. Leaf size=25 \[ \frac {10}{3}+x+\log (x)-\frac {4}{e^{2 x}-x+x^2+\log (x)} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+2 e^{2 x} x \left (3+x^2\right )+2 (1+x) \left (e^{2 x}+(-1+x) x\right ) \log (x)+(1+x) \log ^2(x)}{x \left (e^{2 x}+(-1+x) x+\log (x)\right )^2} \, dx\\ &=\int \left (\frac {1+x}{x}+\frac {8}{e^{2 x}-x+x^2+\log (x)}-\frac {4 \left (-1+x-4 x^2+2 x^3+2 x \log (x)\right )}{x \left (e^{2 x}-x+x^2+\log (x)\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {-1+x-4 x^2+2 x^3+2 x \log (x)}{x \left (e^{2 x}-x+x^2+\log (x)\right )^2} \, dx\right )+8 \int \frac {1}{e^{2 x}-x+x^2+\log (x)} \, dx+\int \frac {1+x}{x} \, dx\\ &=-\left (4 \int \left (\frac {1}{\left (e^{2 x}-x+x^2+\log (x)\right )^2}-\frac {1}{x \left (e^{2 x}-x+x^2+\log (x)\right )^2}-\frac {4 x}{\left (e^{2 x}-x+x^2+\log (x)\right )^2}+\frac {2 x^2}{\left (e^{2 x}-x+x^2+\log (x)\right )^2}+\frac {2 \log (x)}{\left (e^{2 x}-x+x^2+\log (x)\right )^2}\right ) \, dx\right )+8 \int \frac {1}{e^{2 x}-x+x^2+\log (x)} \, dx+\int \left (1+\frac {1}{x}\right ) \, dx\\ &=x+\log (x)-4 \int \frac {1}{\left (e^{2 x}-x+x^2+\log (x)\right )^2} \, dx+4 \int \frac {1}{x \left (e^{2 x}-x+x^2+\log (x)\right )^2} \, dx-8 \int \frac {x^2}{\left (e^{2 x}-x+x^2+\log (x)\right )^2} \, dx-8 \int \frac {\log (x)}{\left (e^{2 x}-x+x^2+\log (x)\right )^2} \, dx+8 \int \frac {1}{e^{2 x}-x+x^2+\log (x)} \, dx+16 \int \frac {x}{\left (e^{2 x}-x+x^2+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 22, normalized size = 0.88 \begin {gather*} x+\log (x)-\frac {4}{e^{2 x}-x+x^2+\log (x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 22, normalized size = 0.88
method | result | size |
risch | \(x +\ln \left (x \right )-\frac {4}{x^{2}+{\mathrm e}^{2 x}+\ln \left (x \right )-x}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 39, normalized size = 1.56 \begin {gather*} \frac {x^{3} - x^{2} + x e^{\left (2 \, x\right )} + x \log \left (x\right ) - 4}{x^{2} - x + e^{\left (2 \, x\right )} + \log \left (x\right )} + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (22) = 44\).
time = 0.35, size = 47, normalized size = 1.88 \begin {gather*} \frac {x^{3} - x^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} + e^{\left (2 \, x\right )}\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4}{x^{2} - x + e^{\left (2 \, x\right )} + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.09, size = 19, normalized size = 0.76 \begin {gather*} x + \log {\left (x \right )} - \frac {4}{x^{2} - x + e^{2 x} + \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (22) = 44\).
time = 0.41, size = 49, normalized size = 1.96 \begin {gather*} \frac {x^{3} + x^{2} \log \left (x\right ) - x^{2} + x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4}{x^{2} - x + e^{\left (2 \, x\right )} + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.11, size = 21, normalized size = 0.84 \begin {gather*} x+\ln \left (x\right )-\frac {4}{{\mathrm {e}}^{2\,x}-x+\ln \left (x\right )+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________